In this paper, we study the maximum density, threshold and emptiness queries for intervals in the streaming model. The input is a stream S of n points in the real line R and a floating closed interval W of width alpha. The specific problems we consider in this paper are as follows. - Maximum density: find a placement of W in R containing the maximum number of points of S. - Threshold query: find a placement of W in R, if it exists, that contains at least Delta elements of S. - Emptiness query: find, if possible, a placement of W within the extent of S so that the interior of W does not contain any element of S. The stream S, being huge, does not fit into main memory and can be read sequentially at most a constant number of times, usually once. The problems studied here in the geometric setting have relations to frequency estimation and heavy hitter identification in a stream of data. We provide lower bounds and results on trade-off between extra space and quality of solution. We also discuss generalizations for the higher dimensional variants for a few cases.
@InProceedings{bishnu_et_al:LIPIcs.FSTTCS.2015.336, author = {Bishnu, Arijit and Chakrabarti, Amit and Nandy, Subhas C. and Sen, Sandeep}, title = {{On Density, Threshold and Emptiness Queries for Intervals in the Streaming Model}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {336--349}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.336}, URN = {urn:nbn:de:0030-drops-56488}, doi = {10.4230/LIPIcs.FSTTCS.2015.336}, annote = {Keywords: Density, threshold, emptiness queries, interval queries, streaming model, heavy hitter, frequency estimation} }
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